where we have denoted \(\omega _{k}\)  by \(\omega \)  for simplicity. Note that one has \(|g_{k}|^{2}/v_{g}=|g_{k}|^{2}(dk/d\omega )=2\pi |g_{\omega }|^{2}D_{\omega }\)  according to the Fermi's golden rule [1]}, [2]}, where \(g_{\omega }\)  is the emitter-waveguide coupling coefficient as a function of \(\omega \)  and \(D_{\omega }\)  is the density of states of the frequency space [3]}, [4]}. Assuming that \(|g(\omega )|^{2}/v_{g}=\gamma /2\)  is constant under the Markovian approximation [5]}, [6]}, [7]} and \(\omega _{0}\)  is far away from the cut-off frequency of the waveguide such that its dispersion can be approximately linearized as \(\omega =\omega _{0}+\nu \)  if  [8]}, [9]}, we have
\(\begin{split}&\frac{dc_{a}(t)}{dt}=-\frac{\gamma }{2\pi }\int _{-\infty }^{+\infty }d\nu \int _{t_{0}}^{t}dt^{\prime }e^{-i\nu (t-t^{\prime })}\lbrace c_{a}(t^{\prime })\\&\qquad \qquad +\cos {[(k_{0}+\frac{\nu }{v_{g}})d]}c_{b}(t^{\prime })\rbrace -iJc_{b}(t),\\&\frac{dc_{b}(t)}{dt}=-\frac{\gamma }{2\pi }\int _{-\infty }^{+\infty }d\nu \int _{t_{0}}^{t}dt^{\prime }e^{-i\nu (t-t^{\prime })}\lbrace c_{b}(t^{\prime })\\&\qquad \qquad +\cos {[(k_{0}+\frac{\nu }{v_{g}})d]}c_{a}(t^{\prime })\rbrace -iJ^{*}c_{a}(t),\end{split}\) 
